Showing papers on "Spectrum of a matrix published in 1969"

Some new bounds on perturbation of subspaces

Abstract: When a Hermitian linear operator A is slightly perturbed, by how much can its invariant subspaces change? Given some approximations to a cluster of neighboring eigenvalues and to the corresponding eigenvectors of a real symmetric matrix, and given a lower bound S > 0 for the gap that separates the cluster from all other eigenvalues, how much can the subspace spanned by the eigenvectors differ from that spanned by our approximations? These questions are closely related; both are investigated here. First the difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other. The angles constitute the spectrum of a Hermitian operator ©, with which is associated a commuting skew-Hermitian operator J=—J; the unitary operator that differs least from the identity and rotates one subspace into the other turns out to be exp(/@). These operators unify the treatment of natural geometric, operatortheoretic and error-analytic questions concerning those subspaces. Given the gap ô, and given bounds upon either the perturbation (1st question) or a computable residual (2nd question), we obtain sharp bounds upon unitary-invariant norms of trigonometric functions of ©. (A norm is unitary-invariant whenever | | i | | = | | C /LFI | for all unitary U and V. Examples are the bound-norm | |L | | I = supj|Lx||/||x|| and the square-norm | | i | | , f f= (trace L*L) .) In this note we consider a finite-dimensional unitary space 3C in which the scalar product is denoted by y*x, and ||x|| = (#*#). Proofs of the following statements will appear elsewhere, together with extensions to infinite-dimensional Hubert spaces and to noncompact or unbounded operators [2]. Tha t article discusses the relation of 'our results to earlier work on the subject, such as [ l ] , [3], [4].

Evaluation of an analytical function of a companion matrix with distinct eigenvalues

Abstract: This letter derives an explicit formula for the evaluation of an arbitrary analytical function of a companion matrix with distinct eigenvalues. The derivation presented here is simple and meaningful. It is also shown herein that this formula has a remarkable recursive property, which makes it extremely attractive for use in digital computations. In fact, the commonly used diagonalization procedure involves, in general, far more computational effort than the use of the procedure derived in this letter.

A theorem on the Lyapunov matrix equation

Abstract: Given the Lyapunov matrix equation A'P + PA + 2\sigmaQ = 0 where σ is some positive scalar, a necessary and sufficient condition for the real parts of the eigenvalues of A to be less than -σ is that P - Q is negative definite. The condition provides an upper bound to the solution of the Lyapunov matrix equation and is useful in the design of minimum-time or minimum-cost linear control systems.